c 2009 by Kyoungsoo Park

250 Pages·2009·15.04 MB·English

(cid:2)c 2009 by Kyoungsoo Park
POTENTIAL-BASED FRACTURE MECHANICS USING COHESIVE ZONE
AND VIRTUAL INTERNAL BOND MODELING
BY
KYOUNGSOO PARK
B.S., Hanyang University, 2003
M.S., University of Illinois at Urbana-Champaign, 2005
DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Civil Engineering
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2009
Urbana, Illinois
Doctoral Committee:
Professor Glaucio H. Paulino, Chair and Director of Research
Associate Professor Jeffery R. Roesler, Co-Director of Research
Assistant Professor C. Armando Duarte
Professor Robert B. Haber
Professor Yonggang Huang, Northwestern University
Associate Professor Karel Matous, University of Notre Dame
Assistant Professor Spandan Maiti, Michigan Technological University
Abstract
The characterization of nonlinear constitutive relationships along fracture surfaces
is a fundamental issue in mixed-mode cohesive fracture simulations. A generalized
potential-based constitutive theory of mixed-mode fracture is proposed in conjunc-
tion with physical quantities such as fracture energy, cohesive strength and shape of
cohesive interactions. The potential-based model is verified and validated by inves-
tigating quasi-static fracture, dynamic fracture, branching and fragmentation. For
quasi-static fracture problems, intrinsic cohesive surface element approaches are uti-
lized to investigate microstructural particle/debonding process within a multiscale
approach. Macroscopic constitutive relationship of materials with microstructure is
estimated by means of an integrated approach involving micromechanics and the
computational model. For dynamic fracture, branching and fragmentation problems,
extrinsic cohesive surface element approaches are employed, which allow adaptive in-
sertion of cohesive surface elements whenever and wherever they are needed. Nodal
perturbation and edge-swap operators are used to reduce mesh bias and to improve
crack path geometry represented by a finite element mesh. Adaptive mesh refinement
and coarsening schemes are systematically developed in conjunction with edge-split
and vertex-removal operators to reduce computational cost. Computational results
demonstrate that the potential-based constitutive model with such adaptive opera-
tors leads to an effective and efficient computational framework to simulate physical
phenomena associated with fracture. In addition, the virtual internal bond model
is utilized for the investigation of quasi-brittle material fracture behavior. All the
computational models have been developed in conjunction with verification and/or
validation procedures.
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To my family
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Acknowledgments
I would like to express my sincere gratitude to my advisor Professor Glaucio H.
Paulino. This thesis would not have been possible without his direction, support and
enthusiasm. I am also indebted to my co-advisor Professor Jeffery R. Roesler for his
invaluable suggestions and encouragement.
I am grateful to Professor Yonggang Huang for his kind advice and insightful
comments, and he has been a source of inspiration. Our collaboration resulted in
Chapter 5 of this thesis and a joint paper. I would like to thank to committee
members, C. Armando Duarte, Robert B. Haber, Karel Matous and Spandan Maiti
for their constructive remarks and suggestions. In addition, I owe many thanks to
Professor Waldemar Celes for his feedback and hands-on help.
I benefited from discussions with my colleagues, Duc Ngo and Rodrigo Espinha,
about micromechanics and topological data structure. I am beholden to my group
members, Matthew C. Walters, Alok Sutradhar, Zhengyu Zhang, Seong-Hyeok Song,
Eshan V. Dave, Bin Shen, Huiming Yin, Shun Wang, Chau H. Le, Tam Nguyen,
Cameron Talischi, Lauren Stromberg, Arun L. Gain, Sofie Leon and Tomas Zegard.
The discussions that we had were enjoyable and valuable.
I can not express enough appreciation to my parents and wonderful sister for their
perpetual support and trust. Finally, I would like to express my deepest thanks to
my lovely wife, Younhee Ko. She has alway been supportive of me, and I treasure
the time that she and I have spent together.
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Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Cohesive Fracture Model . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Cohesive Constitutive Relationships . . . . . . . . . . . . . . . . . . . 3
1.2.1 Non-Potential-Based Models . . . . . . . . . . . . . . . . . . . 3
1.2.2 Potential-Based Models . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Intrinsic versus Extrinsic Cohesive Models . . . . . . . . . . . 6
1.3.2 Enrichment Function Based Approach: GFEM/X-FEM . . . . 8
1.3.3 Finite Elements with Embedded Discontinuities . . . . . . . . 9
1.3.4 Microplane Model . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.5 Atomistic/Continuum Coupling . . . . . . . . . . . . . . . . . 10
1.3.6 Virtual Internal Bond Model . . . . . . . . . . . . . . . . . . . 10
1.3.7 Peridynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.8 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.9 Present Approach . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Research Objective and Thesis Organization . . . . . . . . . . . . . . 12
Chapter 2 Virtual Internal Pair-Bond Model for Quasi-Brittle Ma-
terials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Virtual Internal Bond (VIB) Model Formulation . . . . . . . . . . . . 15
2.2.1 Strain Energy Function in the VIB Model . . . . . . . . . . . 16
2.2.2 Constitutive Relation . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Virtual Bond Density Potential . . . . . . . . . . . . . . . . . 18
2.2.4 Computational Implementation . . . . . . . . . . . . . . . . . 19
2.3 Virtual Internal Pair-Bond (VIPB) Model . . . . . . . . . . . . . . . 19
2.4 Determination of Material Properties . . . . . . . . . . . . . . . . . . 22
2.4.1 Elastic Properties at Infinitesimal Strains . . . . . . . . . . . . 22
2.4.2 Fracture Properties and Mesh Size Dependences . . . . . . . . 23
2.5 Verification – Fracture Properties and Element-Size Dependence . . . 25
2.5.1 Pure Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . 26
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2.5.2 Double Cantilever Beam (DCB) Test . . . . . . . . . . . . . . 27
2.6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 Three-Point Bending (TPB) Tests of Plain Concrete . . . . . 31
2.6.2 On Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.3 TPB Test of Fiber Reinforced Concrete (FRC) . . . . . . . . . 34
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 3 PPR: Unified Potential-Based Cohesive Model of Mixed-
Mode Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Previous Potentials for Cohesive Fracture . . . . . . . . . . . . . . . . 37
3.1.1 Needleman, 1987 . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Needleman, 1990 . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.3 Beltz and Rice, 1991 . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.4 Xu and Needleman, 1993 . . . . . . . . . . . . . . . . . . . . . 44
3.1.5 Limitations of the Exponential Potential . . . . . . . . . . . . 46
3.2 PPR: Unified Potential-Based Constitutive Model . . . . . . . . . . . 48
3.2.1 Definition of the Unified Potential for Mixed-Mode Fracture . 48
3.2.2 Cohesive Interaction (Softening) Region . . . . . . . . . . . . 53
3.2.3 Extension to the Extrinsic Cohesive Zone Model . . . . . . . . 54
3.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Path Dependence of Work-of-Separation . . . . . . . . . . . . . . . . 58
3.3.1 Proportional Separation . . . . . . . . . . . . . . . . . . . . . 58
3.3.2 Non-Proportional Separation . . . . . . . . . . . . . . . . . . . 62
3.4 Mixed-Mode Fracture Verification . . . . . . . . . . . . . . . . . . . . 66
3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter 4 Implementation of the PPR Potential-Based Cohesive
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Topological Data Structure . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 Client-Server Approach . . . . . . . . . . . . . . . . . . . . . . 75
4.3.2 API and Callback functions . . . . . . . . . . . . . . . . . . . 76
4.4 Unloading and Reloading Relationships . . . . . . . . . . . . . . . . . 77
4.4.1 Coupled Unloading/Reloading Model . . . . . . . . . . . . . . 78
4.4.2 Uncoupled Unloading/Reloading Model . . . . . . . . . . . . . 80
4.4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Constitutive Relationships . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.1 Determination of Cohesive Interaction Region . . . . . . . . . 84
4.5.2 Cohesive Traction Vector and Tangent Matrix . . . . . . . . . 84
4.6 Verification of Cohesive Elements . . . . . . . . . . . . . . . . . . . . 88
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Chapter 5 Microstructural Particle/Matrix Debonding Process by
the PPR Potential-Based Model . . . . . . . . . . . . . . . . . . . . 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Constitutive Behavior of Composites with Particle/Matrix Interface
Debonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.1 Hydrostatic Tension Stress State . . . . . . . . . . . . . . . . 94
5.2.2 Extended Mori-Tanaka Method under Hydrostatic Tension . . 95
5.2.3 Extended Mori-Tanaka Method under Equi-biaxial Tension . . 97
5.3 PPR: Potential-Based Cohesive Model for Interface Debonding . . . . 97
5.4 Micromechanics Investigation of the PPR Model . . . . . . . . . . . . 99
5.4.1 Effect of Particle Size . . . . . . . . . . . . . . . . . . . . . . . 100
5.4.2 Effect of Particle Volume Fraction . . . . . . . . . . . . . . . . 101
5.4.3 Effect of Cohesive Energy . . . . . . . . . . . . . . . . . . . . 102
5.4.4 Effect of Cohesive Strength . . . . . . . . . . . . . . . . . . . 103
5.5 Theoretical and Computational Investigation of Materials with Mi-
crostructure Accounting for Particle/Matrix Interface Debonding . . . 104
5.5.1 Particle/Matrix Debonding Process . . . . . . . . . . . . . . . 106
5.5.2 Effect of Microstructure Size . . . . . . . . . . . . . . . . . . . 110
5.5.3 Effect of Particle Elastic Modulus . . . . . . . . . . . . . . . . 111
5.5.4 Effect of Fracture Energy and Particle Size . . . . . . . . . . . 112
5.5.5 Effect of Cohesive Strength and Particle Size . . . . . . . . . . 113
5.5.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.6 Case Study: Determination of the PPR Cohesive Relation . . . . . . 114
5.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Chapter 6 Adaptive Dynamic Cohesive Fracture Simulation Using
Nodal Perturbation and Edge-Swap Operators . . . . . . . . . . . 119
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Toward Unstructured Geometry – Nodal Perturbation (NP) . . . . . 124
6.2.1 Crack Length Convergence . . . . . . . . . . . . . . . . . . . . 126
6.2.2 Crack Angle Convergence . . . . . . . . . . . . . . . . . . . . 129
6.3 Toward Unstructured Topology – Edge Swap (ES) . . . . . . . . . . . 131
6.3.1 Crack Length Convergence . . . . . . . . . . . . . . . . . . . . 131
6.3.2 Crack Angle Convergence . . . . . . . . . . . . . . . . . . . . 133
6.4 Computational Quantification of Isoperimetric Property . . . . . . . . 134
6.5 Nodal Perturbation and Edge Swap Algorithm . . . . . . . . . . . . . 138
6.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.6.1 Compact Compression Specimen (CCS) Test . . . . . . . . . . 140
6.6.2 Microbranching Experiments . . . . . . . . . . . . . . . . . . . 143
6.6.3 Fragmentation Simulations . . . . . . . . . . . . . . . . . . . . 150
6.7 Some Remarks on 4k and Pinwheel Meshes . . . . . . . . . . . . . . 154
6.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
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Chapter 7 Adaptive Mesh Refinement and Coarsening for Cohesive
Dynamic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2.1 Mesh Refinement Schemes . . . . . . . . . . . . . . . . . . . . 159
7.2.2 Refinement Criterion and Interpolation of New Nodes . . . . . 162
7.3 Adaptive Mesh Coarsening . . . . . . . . . . . . . . . . . . . . . . . . 165
7.3.1 Mesh Coarsening Schemes . . . . . . . . . . . . . . . . . . . . 166
7.3.2 Coarsening Criterion and Local Update . . . . . . . . . . . . . 168
7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.4.1 Predefined Crack Path Problem: Mode I Fracture . . . . . . . 170
7.4.2 Mixed-Mode Crack Propagation . . . . . . . . . . . . . . . . . 180
7.4.3 Crack Branching Problem . . . . . . . . . . . . . . . . . . . . 188
7.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Chapter 8 Conclusions and Future Work . . . . . . . . . . . . . . . . 195
8.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 198
Appendix A User-defined element (UEL) subroutine of ABAQUS
for the PPR potential-based cohesive zone model . . . . . . . . . . 201
AppendixB User-definedmaterial(UMAT)subroutineofABAQUS
for the virtual internal pair-bond (VIPB) model . . . . . . . . . . 210
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
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List of Tables
2.1 Relationship between the VIB element size and the fracture energy. . 26
2.2 ElasticandfractureparametersofconcretebeamexperimentsbyRoesler
et al. (2007b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Material properties and the constants in the bond density potential for
each size of beam in the VIB (single-bond) model with the localization
zone size of 0.5 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 The constants in each bond density potential for the VIPB (pair-bond)
model with the localization zone size of 0.5 mm. . . . . . . . . . . . . 33
2.5 Elastic and fracture parameters of plain concrete (Roesler et al., 2007a). 35
3.1 Potentials for cohesive fracture. . . . . . . . . . . . . . . . . . . . . . 38
3.2 Fracture parameters for the unified potential-based model (PPR). . . 58
3.3 Fracture parameters for the model by Xu and Needleman (1993). . . 58
3.4 Geometry of the MMB test specimen. . . . . . . . . . . . . . . . . . . 67
4.1 Comparison between the coupled unloading/reloading and the uncou-
pled unloading/reloading. . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 Properties of high explosive material PBX9501. . . . . . . . . . . . . 118
6.1 Geometrical and topological considerations for cohesive zone model
simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 ρ-path deviation ratio (dev ) and Hausdorff distance (H(p,q)) with
ρ
respect to the number of elements. . . . . . . . . . . . . . . . . . . . 136
7.1 Numbers of nodes and elements, and relative error of the total energy
with respect to the coarsening error levels (e ). . . . . . . . . . . . . 179
4k
7.2 Computational cost comparison for the mixed-mode crack propagation. 184
7.3 Computational cost comparison for the crack branching problem. . . 189
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